bzoj#3456. 城市規劃

題目描述

Solution

用組合意義推很簡單。

$i$個點的簡單無向圖個數為$2^{\left(\genfrac{}{}{0ex}{}{i}{2}\right)}$個。
則其$EGF$為
$$G(x)=\sum _{i>=0}\frac{2^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}}{i!}{x}^i$$
令$i$個點的簡單無向連通圖個數為$f_i$，則其$EGF$為：
$$F(x)=\sum _{i>=0}\frac{f_i}{i!}{x}^i$$
考慮它的含義，簡單無向圖就相當于若干個簡單無向連通圖拼接起來，因此$G(x)=1+\frac{F(x)}{1!}+\frac{F(x{)}^2}{2!}+...$

因此$G(x)=e^{F(x)}$，$F(x)=\ln G(x)$。

因此時間復雜度為$O(nlgn)$

#include <vector> #include <list> #include <map> #include <set> #include <deque> #include <queue> #include <stack> #include <bitset> #include <algorithm> #include <functional> #include <numeric> #include <utility> #include <sstream> #include <iostream> #include <iomanip> #include <cstdio> #include <cmath> #include <cstdlib> #include <cctype> #include <string> #include <cstring> #include <ctime> #include <cassert> #include <string.h> //#include <unordered_set> //#include <unordered_map> //#include <bits/stdc++.h>#define MP(A,B) make_pair(A,B) #define PB(A) push_back(A) #define SIZE(A) ((int)A.size()) #define LEN(A) ((int)A.length()) #define FOR(i,a,b) for(int i=(a);i<(b);++i) #define fi first #define se secondusing namespace std;template<typename T>inline bool upmin(T &x,T y) { return y<x?x=y,1:0; } template<typename T>inline bool upmax(T &x,T y) { return x<y?x=y,1:0; }typedef long long ll; typedef unsigned long long ull; typedef long double lod; typedef pair<int,int> PR; typedef vector<int> VI;const lod eps=1e-11; const lod pi=acos(-1); const int oo=1<<30; const ll loo=1ll<<62; const int mods=1004535809; const int inv2=(mods+1)>>1; const int G=3; const int Gi=(mods+1)/G; const int MAXN=600005; const int INF=0x3f3f3f3f;//1061109567 /*--------------------------------------------------------------------*/ inline int read() {int f=1,x=0; char c=getchar();while (c<'0'||c>'9') { if (c=='-') f=-1; c=getchar(); }while (c>='0'&&c<='9') { x=(x<<3)+(x<<1)+(c^48); c=getchar(); }return x*f; } int a[MAXN],F[MAXN]; namespace Poly {int c[MAXN],f[MAXN],b[MAXN],d[MAXN],rev[MAXN],Limit,l;inline int upd(int x,int y){ return x+y>=mods?x+y-mods:x+y; }inline int quick_pow(int x,int y){int ret=1;for (;y;y>>=1){if (y&1) ret=1ll*ret*x%mods;x=1ll*x*x%mods;}return ret;}void Number_Theoretic_Transform(int *A,int n,int opt){for (int i=0;i<Limit;i++) if (i<rev[i]) swap(A[i],A[rev[i]]);for (int mid=1;mid<Limit;mid<<=1){int Wn=quick_pow((opt==1)?G:Gi,(mods-1)/(mid<<1));for (int j=0;j<Limit;j+=mid<<1)for (int k=j,w=1;k<j+mid;k++,w=1ll*w*Wn%mods){int x=A[k],y=1ll*A[k+mid]*w%mods;A[k]=upd(x,y),A[k+mid]=upd(x,mods-y);}}if (opt==-1){int invLimit=quick_pow(Limit,mods-2);for (int i=0;i<n;i++) A[i]=1ll*A[i]*invLimit%mods;for (int i=n;i<Limit;i++) A[i]=0;}}void getinv(int *F,int *G,int n) //F->invG {if (n==1) { F[0]=quick_pow(G[0],mods-2); return; }getinv(F,G,(n+1)>>1);Limit=1,l=0;while (Limit<(n<<1)) Limit<<=1,l++;for (int i=0;i<Limit;i++) rev[i]=(rev[i>>1]>>1)|((i&1)<<(l-1));for (int i=0;i<n;i++) c[i]=G[i];for (int i=n;i<Limit;i++) c[i]=0;Number_Theoretic_Transform(c,n,1);Number_Theoretic_Transform(F,n,1);for (int i=0;i<Limit;i++) F[i]=1ll*upd(2,mods-1ll*F[i]*c[i]%mods)*F[i]%mods;Number_Theoretic_Transform(F,n,-1);for (int i=n;i<Limit;i++) F[i]=0;}void getln(int *a,int n) //a->ln a (a[0]=1){memset(f,0,sizeof f);getinv(f,a,n);for (int i=0;i<n;i++) a[i]=1ll*a[i+1]*(i+1)%mods;Limit=1,l=0;while (Limit<(n<<1)) Limit<<=1,l++;for (int i=0;i<Limit;i++) rev[i]=(rev[i>>1]>>1)|((i&1)<<(l-1));Number_Theoretic_Transform(a,n,1);Number_Theoretic_Transform(f,n,1);for (int i=0;i<Limit;i++) f[i]=1ll*f[i]*a[i]%mods;Number_Theoretic_Transform(f,n,-1);for (int i=1;i<n;i++) a[i]=1ll*f[i-1]*quick_pow(i,mods-2)%mods; for (int i=n;i<=Limit;i++) a[i]=0; a[0]=0;}void getexp(int *F,int *a,int n) //F->exp a (a[0]=0){memset(b,0,sizeof b);if (n==1) { F[0]=1; return; }getexp(F,a,(n+1)>>1);for (int i=0;i<n;i++) b[i]=F[i];getln(b,n);Limit=1,l=0;while (Limit<(n<<1)) Limit<<=1,l++;for (int i=0;i<Limit;i++) rev[i]=(rev[i>>1]>>1)|((i&1)<<(l-1));for (int i=0;i<n;i++) b[i]=upd(a[i],mods-b[i]);for (int i=n;i<Limit;i++) b[i]=F[i]=0;b[0]++;Number_Theoretic_Transform(F,n,1);Number_Theoretic_Transform(b,n,1);for (int i=0;i<Limit;i++) F[i]=1ll*F[i]*b[i]%mods;Number_Theoretic_Transform(F,n,-1);for (int i=n;i<Limit;i++) F[i]=0; }void getsqrt(int *F,int *a,int n,int k) //F->sqrt^k a(a[0]=1){memset(d,0,sizeof d);for (int i=0;i<n;i++) d[i]=a[i];getln(d,n);int invk=quick_pow(k,mods-2);for (int i=0;i<n;i++) d[i]=1ll*d[i]*invk%mods;getexp(F,d,n);} void getpow(int *F,int *a,int n,int k) //F->pow^k a(a[0]=1){memset(d,0,sizeof d);for (int i=0;i<n;i++) d[i]=a[i];getln(d,n);for (int i=0;i<n;i++) d[i]=1ll*d[i]*k%mods;getexp(F,d,n);} } int fac[MAXN],inv[MAXN]; void Init(int n) {fac[0]=1;for (int i=1;i<=n;i++) fac[i]=1ll*fac[i-1]*i%mods;inv[n]=Poly::quick_pow(fac[n],mods-2);for (int i=n-1;i>=0;i--) inv[i]=1ll*inv[i+1]*(i+1)%mods; } int main() {int n=read();Init(n+5);for (int i=0;i<=n;i++) a[i]=1ll*Poly::quick_pow(2,1ll*i*(i-1)/2%(mods-1))*inv[i]%mods; // for (int i=0;i<=n;i++) cout<<i<<":"<<a[i]<<endl;Poly::getln(a,n+1); // for (int i=0;i<=n;i++) cout<<i<<":"<<a[i]<<endl;printf("%d\n",1ll*a[n]*fac[n]%mods);return 0; }

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